Nonlinear transport and dynamics of fluctuations

The time evolution of the macroscopic variables of a system initially in a state far from thermal equilibrium is studied from a statistical mechanical point of view. Exact nonlinear transport equations for the mean values and linear nonstationary Langevin equations for the fluctuations around the mean path are derived. Connections between the dynamics of fluctuations and the transport equations are discussed. The Langevin random forces depend on the macroscopic state and they are related to the transport kernels by a fluctuation-dissipation formula.     

Classical and quantum kinetic equations with exact conservation laws

Stationary and dynamic properties of reduced density matrices can be determined from formal or approximate closures of an infinite hierarchy of equations. The local macroscopic conservation laws place weak but important constraints on the reduced density matrices which should be respected by any closure. For pairwise additive forces conditions on the closure of the one- and two-particle equations are obtained that preserve the exact functional dependence of the conserved densities and their fluxes on the reduced density matrices. To illustrate the nature of these conditions, a closure approximation suitable for a quantum gas is given, yielding an extension of the time-dependent Hartree-Fock equations for the dynamics of a nuclear fluid to include collisions.     

Continuous and Discrete Adjoint Approach Based on Lattice Boltzmann Method in Aerodynamic Optimization Part I: Mathematical Derivation of Adjoint Lattice Boltzmann Equations

The significance of flow optimization utilizing the lattice Boltzmann (LB) method becomes obvious regarding its advantages as a novel flow field solution method compared to the other conventional computational fluid dynamics techniques. These unique characteristics of the LB method form the main idea of its application to optimization problems. In this research, for the first time, both continuous and discrete adjoint equations were extracted based on the LB method using a general procedure with low implementation cost. The proposed approach could be performed similarly for any optimization problem with the corresponding cost function and design variables vector. Moreover, this approach was not limited to flow fields and could be employed for steady as well as unsteady flows. Initially, the continuous and discrete adjoint LB equations and the cost function gradient vector were derived mathematically in detail using the continuous and discrete LB equations in space and time, respectively. Meanwhile, new adjoint concepts in lattice space were introduced. Finally, the analytical evaluation of the adjoint distribution functions and the cost function gradients was carried out.     

Evolution of the Distribution of Wealth in an Economic Environment Driven by Local Nash Equilibria

We present and analyze a model for the evolution of the wealth distribution within a heterogeneous economic environment. The model considers a system of rational agents interacting in a game theoretical framework, through fairly general assumptions on the cost function. This evolution drives the dynamic of the agents in both wealth and economic configuration variables. We consider a regime of scale separation where the large scale dynamics is given by a hydrodynamic closure with a Nash equilibrium serving as the local thermodynamic equilibrium. The result is a system of gas dynamics-type equations for the density and average wealth of the agents on large scales. We recover the inverse gamma distribution as an equilibrium in the particular case of quadratic cost functions which has been previously considered in the literature.     

湍流两相流动有燃烧颗粒相概率密度函数输运方程理论

由有燃烧的湍流气粒两相流动的瞬态方程和统计力学概率密度函数概念出发,推导了有燃烧颗粒相的质量－动量－能量联合概率密度函数（ＰＤＦ）输运方程,并对方程中条件期望项用梯度模拟概念进行了模拟封闭。封闭后的ＰＤＦ方程可作为建立颗粒拟流体模型方程和封闭二阶矩模型的基础,也可以通过Ｍｏｎｔｅ－Ｃａｒｌｏ 法求解用以直接计算颗粒雷诺应力和湍流动能,以便和二阶 矩模型的结果相对照,改善二阶矩模型。     

ON AN APPROXIMATE FORM OF THE COUPLED EQUATIONS OF THE ORDER PARAMETER WITH THE WEAK ELECTROMAGNETIC FIELD FOR THE IDEAL SUPERCONDUCTOR

A simplified derivation of the macroscopic electrodynamic equations of Umezawa, Hancini et al. for superconductors is given in the framework of the closed time path Green's functions (CTPGF)using generalized Ward-Takahashi identities. It is shown that the forms of the equations obtained are the same for both thermoe quilibrium and nonequilibrium stationary states provided the electromagnetic field is weak and its effect on the modulus of the order parameter can be neglected. The statistical behavior of the states is completely specified in the equations by parameters which can be calculated by the method of CTPGF.     

Nonlinear-closed equations for the first and the second moments of macroscopic variables

A quantum-mechanical theory of describing systems far from equilibrium is developed. A set of time evolution equations for every moment of macroscopic variables is derived with the aid of the new idempotent operator. From this set of equations nonlinear but closed equations for the first and the second moments are obtained directly. The theory is applied to the problem of a spin interacting with its surroundings. The Bloch equation with the Landau-Lifshitz friction term is derived quantum mechanically. The relation between this method and that of system size expansion is discussed.     

Fluctuations in nonlinear Markovian systems

We study nonlinear irreversible processes by statistical mechanical methods from a general point of view. Assuming that the macroscopic variables behave approximatively Markovian we derive evolution equations for the mean values as well as for the fluctuations about the mean. The mean values obey nonlinear transport equations and the fluctuations obey linear nonstationary Langevin equations. The equations of motion are completely specified by the entropy and the transport coefficients as functions of the macroscopic state. The theory provides a statistical mechanical basis for some phenomenological approaches put forward recently.     

Quasi-equilibrium closure hierarchies for the Boltzmann equation

In this paper, explicit method of constructing approximations (the triangle entropy method) is developed for nonequilibrium problems. This method enables one to treat any complicated nonlinear functionals that fit best the physics of a problem (such as, for example, rates of processes) as new independent variables.

The work of the method is demonstrated on the Boltzmann's-type kinetics. New macroscopic variables are introduced (moments of the Boltzmann collision integral, or scattering rates). They are treated as independent variables rather than as infinite moment series. This approach gives the complete account of rates of scattering processes. Transport equations for scattering rates are obtained (the second hydrodynamic chain), similar to the usual moment chain (the first hydrodynamic chain). Various examples of the closure of the first, of the second, and of the mixed hydrodynamic chains are considered for the hard sphere model. It is shown, in particular, that the complete account of scattering processes leads to a renormalization of transport coefficients.

The method gives the explicit solution for the closure problem, provides thermodynamic properties of reduced models, and can be applied to any kinetic equation with a thermodynamic Lyapunov function.     

von Kármán–Howarth and Corrsin equations closure based on Lagrangian description of the fluid motion

A new approach to obtain the closure formulas for the von Kármán–Howarth and Corrsin equations is presented, which is based on the Lagrangian representation of the fluid motion, and on the Liouville theorem associated to the kinematics of a pair of fluid particles. This kinematics is characterized by the finite scale separation vector which is assumed to be statistically independent from the velocity field. Such assumption is justified by the hypothesis of fully developed turbulence and by the property that this vector varies much more rapidly than the velocity field. This formulation leads to the closure formulas of von Kármán–Howarth and Corrsin equations in terms of longitudinal velocity and temperature correlations following a demonstration completely different with respect to the previous works. Some of the properties and the limitations of the closed equations are discussed. In particular, we show that the times of evolution of the developed kinetic energy and temperature spectra are finite quantities which depend on the initial conditions.     

Basic concepts of synergetics

The paper presents a brief outline of microscopic as well as of macroscopic synergetics. In microscopic synergetics we start from evolution equations for microscopic variables or densities in which fluctuating forces and control parameters are included. When control parameters are changed, the systems are studied close to instability points. The concepts of order parameters, enslaving, critical fluctuations, and critical slowing down are presented. In macroscopic synergetics unbiased estimates on distribution functions and underlying processes are made based on observed moments or correlation functions. In such a case, a Fokker-Planck equation or a corresponding Langevin equation may be derived.     

A Hopf-like equation and perturbation theory for differential delay equations

We extend techniques developed for the study of turbulent fluid flows to the statistical study of the dynamics of differential delay equations. Because the phase spaces of differential delay equations are infinite dimensional, phase-space densities for these systems are functionals. We derive a Hopf-like functional differential equation governing the evolution of these densities. The functional differential equation is reduced to an infinite chain of linear partial differential equations using perturbation theory. A necessary condition for a measure to be invariant under the action of a nonlinear differential delay equation is given. Finally, we show that the evolution equation for the density functional is the Fourier transform of the infinite-dimensional version of the Kramers-Moyal expansion.     

Reduced Order Podolsky Model

We perform the canonical and path integral quantizations of a lower-order derivatives model describing Podolsky’s generalized electrodynamics. The physical content of the model shows an auxiliary massive vector field coupled to the usual electromagnetic field. The equivalence with Podolsky’s original model is studied at classical and quantum levels. Concerning the dynamical time evolution, we obtain a theory with two first-class and two second-class constraints in phase space. We calculate explicitly the corresponding Dirac brackets involving both vector fields. We use the Senjanovic procedure to implement the second-class constraints and the Batalin-Fradkin-Vilkovisky path integral quantization scheme to deal with the symmetries generated by the first-class constraints. The physical interpretation of the results turns out to be simpler due to the reduced derivatives order permeating the equations of motion, Dirac brackets and effective action.     

Macroscopic variables

The problem of describing macroscopic variables in quantum theory, is discussed. It is suggested that the eigenspaces of macroscopic variables be hyperspheres rather than closed linear subspaces. This is combined with the usual suggestion that macroscopic variables are nearly diagonal in the energy representation. The Schrödinger paradox is resolved in terms of this discussion.     

Fierz bilinear formulation of the Maxwell–Dirac equations and symmetry reductions

We study the Maxwell–Dirac equations in a manifestly gauge invariant presentation using only the spinor bilinear scalar and pseudoscalar densities, and the vector and pseudovector currents, together with their quadratic Fierz relations. The internally produced vector potential is expressed via algebraic manipulation of the Dirac equation, as a rational function of the Fierz bilinears and first derivatives (valid on the support of the scalar density), which allows a gauge invariant vector potential to be defined. This leads to a Fierz bilinear formulation of the Maxwell tensor and of the Maxwell–Dirac equations, without any reference to gauge dependent quantities. We show how demanding invariance of tensor fields under the action of a fixed (but arbitrary) Lie subgroup of the Poincaré group leads to symmetry reduced equations. The procedure is illustrated, and the reduced equations worked out explicitly for standard spherical and cylindrical cases, which are coupled third order nonlinear PDEs. Spherical symmetry necessitates the existence of magnetic monopoles, which do not affect the coupled Maxwell–Dirac system due to magnetic terms cancelling. In this paper we do not take up numerical computations. As a demonstration of the power of our approach, we also work out the symmetry reduced equations for two distinct classes of dimension 4 one-parameter families of Poincaré subgroups, one splitting and one non-splitting. The splitting class yields no solutions, whereas for the non-splitting class we find a family of formal exact solutions in closed form.     

Microstructural instabilities: dislocation substructures with pronounced mechanical effects

Microstructure evolution is largely dominated by the internal stress fields that appear upon the appearance of inhomogeneous structures in a material. The hardening behaviour of metals physically originates from such a complex microstructure evolution. As deformation proceeds, statistically homogeneous distributions of dislocations in grains become unstable, which constitutes the driving force for the development of a pronounced dislocation substructure. The dislocation structure already appears at early stages of deformation due to the statistical trapping of dislocations. Cell walls contain dislocation dipoles and multipoles with high dislocation densities and enclose cell-interior regions with a considerably smaller dislocation density. The presence and evolution of such a dislocation arrangement in the material influence the mechanical response of the material and is commonly associated with the transient hardening after strain path changes. This contribution introduces a micromechanical continuum model of the dislocation cell structure based on the physics of the dislocation interactions. The approximation of the internal stress field in such a microstructure and the impact on the macroscopic mechanical response are the main items investigated here.     

Data-Driven Model Reduction for Stochastic Burgers Equations

We present a class of efficient parametric closure models for 1D stochastic Burgers equations. Casting it as statistical learning of the flow map, we derive the parametric form by representing the unresolved high wavenumber Fourier modes as functionals of the resolved variable’s trajectory. The reduced models are nonlinear autoregression (NAR) time series models, with coefficients estimated from data by least squares. The NAR models can accurately reproduce the energy spectrum, the invariant densities, and the autocorrelations. Taking advantage of the simplicity of the NAR models, we investigate maximal space-time reduction. Reduction in space dimension is unlimited, and NAR models with two Fourier modes can perform well. The NAR model’s stability limits time reduction, with a maximal time step smaller than that of the K-mode Galerkin system. We report a potential criterion for optimal space-time reduction: the NAR models achieve minimal relative error in the energy spectrum at the time step, where the K-mode Galerkin system’s mean Courant–Friedrichs–Lewy (CFL) number agrees with that of the full model.